round the answers for part c to the nearest cent

### (c) Find the average cost and the marginal average cost for a production level of 25 units, rounded to the nearest cent.

Let's calculate $AC(25)$ and $MAC(25)$ again, rounding to the nearest cent:

#### Average Cost:
\[ AC(25) = \frac{3(25) - 1}{8(25)^2 + 9(25)} \]
\[ AC(25) = \frac{75 - 1}{8 \cdot 625 + 225} \]
\[ AC(25) = \frac{74}{5000 + 225} \]
\[ AC(25) = \frac{74}{5225} \]
\[ AC(25) \approx 0.0142 \text{ (in hundreds of dollars)} \]

Converting to dollars and rounding to the nearest cent:
\[ AC(25) \approx 0.0142 \times 100 = 1.42 \text{ dollars} \]

#### Marginal Average Cost:
\[ MAC(25) = \frac{-24(25)^2 + 16(25) - 9}{(8(25)^2 + 9(25))^2} \]
\[ MAC(25) = \frac{-24 \cdot 625 + 400 - 9}{(5000 + 225)^2} \]
\[ MAC(25) = \frac{-15000 + 400 - 9}{5225^2} \]
\[ MAC(25) = \frac{-14609}{27350625} \]
\[ MAC(25) \approx -0.000534 \text{ (in hundreds of dollars)} \]

Converting to dollars and rounding to the nearest cent:
\[ MAC(25) \approx -0.000534 \times 100 = -0.0534 \text{ dollars} \]
\[ MAC(25) \approx -0.05 \text{ dollars} \]

### Interpretation:

- The average cost for producing 25 units of perfume is approximately $1.42 per unit.
- The marginal average cost for producing 25 units of perfume is approximately -$0.05 per additional unit, indicating a slight decrease in the average cost as more units are produced.

Do you want more details or have any questions? Here are some questions you might find helpful:

1. Can you explain the quotient rule in more detail?
2. How would the cost functions change if the total cost function were different?
3. What does the negative marginal average cost indicate in a real-world scenario?
4. Can you show the detailed steps of the differentiation using the quotient rule?
5. How can these cost functions be applied to optimize production levels?

**Tip:** When interpreting cost functions, consider both the mathematical results and their practical implications in terms of production efficiency and cost management.

Discover how to calculate the average cost and marginal average cost for producing 25 units. The average cost is approximately $1.42 per unit, while the marginal average cost is about -$0.05 per additional unit, reflecting cost behavior in production. Ideal for advanced high school students.

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